Per-Capita Rates — AP® Environmental Science Notes

Environmental data comes as totals: a city uses 4.5 million liters of water a day, a country emits 5 billion metric tons of carbon dioxide a year. Totals alone cannot tell you whether a place uses resources heavily, because a big total might just mean many people. Rates fix that. This chapter shows how to build a rate, how to read "per", and how to keep the units straight.

What a rate is

A rate compares two quantities measured in different units by dividing one by the other: liters per day, kilometers per hour, people per square kilometer. A unit rate is the rate for exactly one of the second quantity: 150 liters per 1 person, written 150 L/person.

The word "per" means "divided by", and the quantity after it goes on the bottom. "Liters per kilometer" is $\text{L} \div \text{km}$ . "Kilometers per liter" is $\text{km} \div \text{L}$ . They are different numbers, so read the question twice.

Per capita

Per capita means per person. Divide a total by the population.

Two bar charts from invented data. Total use: Town A uses 4.5 million liters a day and Town B 2 million liters a day, so Town A uses more in total. Use per person: Town A has 30,000 people, so 4,500,000 divided by 30,000 is 150 liters per person per day; Town B has 8,000 people, so 2,000,000 divided by 8,000 is 250 liters per person per day. Town B uses more per person. — **Figure 1.** Town A uses more water in total, but Town B uses more per person. LevlPrep original diagram.

Worked example. Town A has 30,000 people and uses 4,500,000 L of water a day. Town B has 8,000 people and uses 2,000,000 L a day. Which uses more per person?

Town A setup: $\frac{4{,}500{,}000\,\text{L}}{30{,}000\,\text{people}}$ $= 150$ L per person per day.

Town B setup: $\frac{2{,}000{,}000\,\text{L}}{8{,}000\,\text{people}}$ $= 250$ L per person per day.

Town A uses more in total; Town B uses more per person. The per-capita value is the fair comparison.

Density and rates over time

Kind of rate	Divide	Example
Per capita	total ÷ people	4,500,000 L ÷ 30,000 people = 150 L/person
Density	count ÷ area	400,000 people ÷ 2,500 km² = 160 people/km²
Rate of change	change ÷ time	36 cm ÷ 12 weeks = 3 cm/week
Use per distance	amount ÷ distance	1,200 L ÷ 15,000 km = 0.08 L/km

Write the setup with units

On the exam, the setup earns its own point. Write the numbers with their units as a fraction, then the answer with its unit:

Worked example. A car used 1,200 L of gasoline in 15,000 km. Find the fuel used per km.

Setup: $\frac{1{,}200\,\text{L}}{15{,}000\,\text{km}}$

Answer: $0.08$ L/km. If you had divided the other way you would get 12.5 km/L, a correct number for a different question; the units tell you which one you found.

If the numbers come in different units, convert first. A school of 800 students wastes 2.4 metric tons of food a month. Change 2.4 metric tons to 2,400 kg, then $\frac{2{,}400\,\text{kg}}{800\,\text{students}}$ $= 3$ kg per student per month.

Check that it makes sense

Per-person water use in a town should be tens to a few hundred liters a day, not millions.
If your answer is tiny when you expected something large, you probably divided upside down.
Totals reflect size; rates reflect behavior. A headline that calls the biggest city "the most wasteful" may be wrong once you divide by population.